English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

We establish uniform error bounds of an exponential wave integrator Fourierpseudospectral (EWI-FP) method for the long-time dynamics of the nonlinearKlein-Gordon equation (NKGE) with a cubic nonlinearity whose strength ischaracterized by $\varepsilon^2$ with $\varepsilon \in (0, 1]$ a dimensionlessparameter. When $0 < \varepsilon \ll 1$, the problem is equivalent to thelong-time dynamics of the NKGE with small initial data (and $O(1)$ cubicnonlinearity), while the amplitude of the initial data (and the solution) is at$O(\varepsilon)$. For the long-time dynamics of the NKGE up to the time at$O(1/\varepsilon^{2})$, the resolution and error bounds of the classicalnumerical methods depend significantly on the small parameter $\varepsilon$,which causes severe numerical burdens as $\varepsilon \to 0^+$. The EWI-FPmethod is fully explicit, symmetric in time and has many superior properties insolving wave equations. By adapting the energy method combined with the methodof mathematical induction, we rigorously carry out the uniform error bounds ofthe EWI-FP discretization at $O(h^{m_0} + \varepsilon^{2-\beta}\tau^2)$ up tothe time at $O(1/\varepsilon^{\beta})$ with $0 \leq \beta \leq 2$, mesh size$h$, time step $\tau$ and $m_0$ an integer depending on the regularity of thesolution. By a rescaling in time, our results are straightforwardly extended tothe error bounds and $\varepsilon$-scalability (or meshing strategyrequirement) of the EWI-FP method for an oscillatory NKGE, whose solutionpropagates waves with wavelength at $O(1)$ and $O(\varepsilon^{\beta})$ inspace and time, respectively, and wave speed at $O(\varepsilon^{-\beta})$.Finally, extensive numerical results are reported to confirm our errorestimates.

Uniform error bounds of an exponential wave integrator for the long-time dynamics of the nonlinear Klein-Gordon equation